  The intercept form of a quadratic equation is as follows:

`y = a(x - p)(x - q)`

 The intercept form of a quadratic equation is the form of a quadratic equation by which you can easily tell the x intercepts of the quadratic equation.
It is thus very useful to convert a quadratic equation to its intercept form in order to get the value of its two x-intercepts.

Meaning of the various symbols in the intercept form of a quadratic equation:

• `p` and `q` are the x-intercepts (also zeros, solutions, roots)
• The axis of symmetry can be found out from the values of `p` and `q` by the equation  `x = (p + q)/2`. This is because the axis of symmetry passes through the vertex and hence is in the exact middle of the two x-intercepts. If there is only one x-intercept, or if both x-intercepts are same, then the axis of symmetry is equal to `x = p = q`.
• `a` is the vertical stretching factor
• `a` can't be equal to zero, because then the equation will not be a quadratic one
• If `a` is negative, the parabola faces downwards, and if `a` is positive then the parabola faces upwards
• If the absolute value of `a`, that is `|a|` is greater than 1 then the parabola gets narrower because it gets stretched vertically and if it is lesser than 1 then the parabola gets wider because it gets stretched vertically by a fractional factor whose denominator is greater than the numerator.
The x-intercepts of parabolas can be obtained from their quadratic equations in intercept forms as follows:

Quadratic Equation in intercept formX-intercepts of its parabola
y = (x - 3)(x - 4)3 and 4
y = x(x - 5)0 and 5
y = (x + 3)(x + 5)-3 and -5

It is also very easy to get the equation of the axis of symmetry of a parabola from the intercept form of a quadratic equation. The axis of symmetry can be calculated with the help of the following formula:
Axis of symmetry, `x = (p + q)/2`
In the above formula, p and q get their values from the intercept form of the quadratic equation.

For example, for the quadratic equation `y = (x - 3)(x - 4)`, the axis of symmetry is :
`x = (3 + 4)/2 = 7/2`

So far, we discussed the quadratics in the form f(x) = ax2 + bx + c, where a ≠ 0. This form is called standard from. If a quadratic function is in the form f(x) = a(xh)2 + k, then it is called vertex form. In this function, the point (h, k) is the vertex. For example, if f(x) = 5(x – 3)2 – 11, then the vertex is at the point (3, –11).

Another form of a quadratic function is called intercept form. This form of the function is f(x) = a(xp)(x q). The points x = p and x = q are the x-intercepts. Any function in intercept form can be transformed to standard form by multiplying the parentheses. But a function in standard form might not be possible to be transformed to the intercept form.

For example, the function f(x)= 3(x – 4)(x + 1) can be transformed to a standard form as shown below:
f(x) = 3(x – 4)(x + 1)

Multiply the right side using FOIL method. Practice 4. Change the following functions to standard form.

(a) f(x) = 3(x + 11)2 – 5  Vertex Form
(b) f(x) = 2(x – 1)(x + 7)  Intercept Form

Solution
(a) Expand the parentheses first.

f(x) = 3(x2 + 22x + 121) – 5

Distribute 3 over the parentheses. Combine the like terms. (b) Multiply the parentheses on the right using the FOIL method.

f(x) = 2(x2 + 6x – 7)

Distribute 2 over the parentheses.

f(x)  = 2x2 + 12x – 14 Standard Form

Video Instruction
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Now let's do Practice Exercise 5-3 (top).

Sours: https://legacy.etap.org/demo/algebra2/lesson5/instruction3tutor.html

## -intercepts of a Quadratic Function

The -intercepts of the function are the solutions of the quadratic equation . The solutions of a quadratic equation of the form are given by the quadratic formula.

Here, the expression is called the discriminant.

The discriminant can be used to confirm the number of -intercepts and the type of solutions of the quadratic equation.

 Value of Discriminant Type and Number of -intercepts (roots) Example of Graph of Related Function is a perfect square real and rational is not a perfect square real and irrational real No -intercepts Graphing Parabolas in Intercept Form

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